Inertia term

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

Step 1 To solve a Stokes flow problem by this program the inertia term in the elemental stiffness matrix should be eliminated. Multiplication of the density variable by zero enforces this conversion (this variable is identified in the program listing). 

Fig. 3-11 shows that, foi watei, entropy and heat capacity ai e summations in which two terms dominate, the translational energy of motion of molecules treated as ideal gas paiticles. and rotational, energy of spin about axes having nonzero rnorncuts of inertia terms (see Prublerris). 

In creeping flow with the inertia term neglected, the velocity distribution rapidly reaches a steady value after a distance of r0 inside a capillary tube. At this stage the velocity distribution showed the typical parabolic shape characteristic of a Poiseuille flow. In the case of inviscid flow where inertia is the predominant term, it takes typically (depending on the Reynolds number) a distance of 20 to 50 diameters for the flow to be fully developed (Fig. 34). With the short capillary section ( 4r0) in the present design, the velocity front remains essentially unperturbed and the velocity along the symmetry axis, i.e. vx (y = 0), is identical to v0. 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

Because of nonlinear Interactions between buoyancy, viscous and Inertia terms multiple stable flow fields may exist for the same parameter values as also predicted by Kusumoto et al (M.). The bifurcations underlying this phenomenon may be computed by the techniques described In the numerical analysis section. The solution structure Is Illustrated In Figure 7 In terms of the Nusselt number (Nu, a measure of the growth rate) for varying Inlet flow rate and susceptor temperature. Here the Nusselt number Is defined as ... 

The physical significance of the first term is obvious. The second term is the inertia term. The third term accounts for gravitational effects it vanishes for horizontal pipelines. The last term represents the frictional losses. 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

A AZPE = AZPEii — AZPEd AZPEt) corresponds to the terms for the reactions of monodeuteriated aldehydes. Terms defined by IE = MMl x EXC x EXP (IE is the Isotopic exchange equilibrium, MMl is the mass moment of inertia term representing the rotational and translational partition function ratios, EXC is the vibrational excitation term and EXP is the exponential zero point energy). 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

Mason and co-workers (B8, F3, Gll, M5, T15) have shown that Eqs. (10-32) to (10-35) can also be applied to disks and cylinders provided that one uses an apparent value of , calculated from Eq. (10-36) and the observed Bretherton (B15) considered more general shapes and proved that most bodies of revolution, except for some extreme shapes, show periodic rotation with no lateral migration (i.e., no lift) provided that inertia terms are neglected. In reality all these particles migrate in the direction of positive lift (see Chapter 9). For a useful extended review on particle motion in shear fields, see Goldsmith and Mason (G12). 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

Johnson proposed a more general set of linear dynamic equations taking into account a mass coupling in the inertia terms [73]. This theory is based on a general phenomenological scheme originally proposed by Biot to describe dynamics in porous media [74,75], Johnson claimed that the inertia terms, the left hand sides of Eqs. (6.1) and (6.2), should in principle be replaced by... 

The following approximations are made DL SD8 (the substantial derivative) mdl y/dB (ignoring flux and inertia terms) AirDp pp/S (nondeformable sphere) and X>0 (in the horizontal dimension) and = AfplPp—Pi)g/pp (>n the vertical dimension). 

Semenov (S7) simplified the wavy flow equations by omitting the inertia terms, which is permissible in the case of very thin films. Expressions are obtained for the wavelength, wave velocity, surface shape, stability, etc., with an adjoining gas stream the treatment refers mainly to the case of upward cocurrent flow of the gas and wavy film in a vertical tube. 

Semenov (S7), 1950 Extension of earlier work to wavy film flow. Kapitsa theory simplified by omitting inertia terms, and applied to wavy film flow with co- or counter-flow of gas to give thickness, velocity, wavelength, wave velocity, stability, onset of flooding, etc. 

In practice, the recirculation cells are often eliminated by increasing the inlet flow rate, a procedure that also improves film thickness uniformity. However, because of nonlinear interactions among buoyancy, viscosity, and inertia terms, the transitions between the flow patterns may be abrupt, and multiple stable-flow fields may exist for the same parameter values (24,195,... 

In addition, the continuity equation also tells us that the two inertia terms in the x-momentum equation are of similar magnitude, i.e.,... 

First, consider the case where the flow is parallel to the cylinders. It is assumed that the fluid is moving through the annular space between the cylinder of radius a and the fluid envelope of equivalent radius b, as shown in Fig. 7.14. Assume that the fluid motion is in the creeping flow regime so that inertia terms can be omitted from the Navier-Stokes equations. Thus, in cylindrical coordinates, we have... 

It is assumed that the EOF is generated only at the charged wall and the packing particles are uncharged. The flow can then be visualized in the form of very thin annuli of liquid in the packed column. Each annulus faces a force in the forward direction (the direction of EOF) from the annulus enveloping it and a force in the backward direction from the annulus inside it. The inertia terms and the compressibility of the fluid are assumed to be negligibly small. The net viscous force, Fv, on such an annulus of unit volume in absence of any particles is given by... 

Limitations imposed by slip and inertia terms upon Stokes law for the motion of spheres through liquids. Phil. Mag., 22 (6th Ser.) 755-775. Arrhenius, Svante... 

If the Prandtl number is large, the first term on the right-hand side of this equation, i.e., the inertia term, will be negligible, i.e., the equation will effectively be ... 

It will be seen from the results given in Fig. E8.8 that Nu does appear to be essentially independent of Pr when Pr > 2. It will also be seen that the change in Nu over the entire Pr range considered is small. This indicates that the inertia term in the vorticity equation, i.e. ... 

It will be seen from the results given by the similarity solution that the velocities are very low in natural convective boundary layers in fluids with high Prandtl numbers. In such circumstances, the inertia terms (i.e., the convective terms) in the momentum equation are negligible and the boundary layer momentum equation for a vertical surface effectively is ... 

While it is easy to extend the governing equations to include the inertia terms, these terms are seldom important in studying the flow through a porous medium. 

Both in hydrodynamics and in chemical kinetics, instability may occur due to nonlinear conditions far from equilibrium. In hydrodynamic systems, nonlinear conditions are produced by the inertia terms, such as the critical Reynolds number or Rayleigh number. However, nonequilibrium kinetic conditions may lead to a variety of structures. In chemical systems, some autocatalytic effect is required for instability. 

Eq. (4.16). Ho %ver, this result is unphysical for long rods. Let a cylindrical gel have a length L and a radius R with L> R. Then Eq. (4.30) implies flo whereas the correct answer is flo K/R as will be derived by accounting for the fluid velocity fidd in Section 6. On the other hand, Eq. (4.23) will remain correct for spherical diapes. We note that the above theory including Eq. (4.30) can be used for solid materials near the point K = 0. Since C is very small in solids [36], we retain the inertia term in Eq. (4.15) to obtain an equation for Qq valid for R <... 

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